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From: Mike Dougherty on 25 Dec 2007 00:14
Something my little pea brain has never understood is why, going back
to the early '80's, the HP 15c was able to preform numerical
integration, but not numeric differention.

Q1: Does anyone know why??

Q2: Is it within the capibilities of these calculators to find the
numeric derivative of a function, given the proper paramteters?

Q3: Has anyone written a program to perform that operation, and made
it available? Where?
From: Dieter on 25 Dec 2007 18:32
Mike Dougherty wrote:

> Something my little pea brain has never understood is why, going back
> to the early '80's, the HP 15c was able to preform numerical
> integration, but not numeric differention.
>
> Q1: Does anyone know why??

Probably because it's so trivial that HP didn't bother to integrate it.
Into the 15C, that is. 8-)

> Q2: Is it within the capibilities of these calculators to find the
> numeric derivative of a function, given the proper paramteters?
>
> Q3: Has anyone written a program to perform that operation, and made
> it available? Where?

It's so simple that probably noone would publish it as a proof of his
programming skills:

As you might know, f'(x) = lim (h->0) of (f(x)-f(x+h))/h

For numerical differentiation this expression is used with a small
(but not too small) h in order to approximate the limit, i.e. the
first derivate of f.

Somewhat better precision may be obtained by evaluating the term
(f(x+h)-f(x-h))/2h instead.

h can be provided by the user, or the program itself can set an
appropriate value. A good starting point is h = x/10000 (for x<>0).
The smaller h gets, the better the approximation. OTOH if h gets too
small, the calculator might not be able to distinguish f(x+h) from
f(x-h). Which is just *one* problem here. <8)

Enter f(x) the same way you'd do for integration: use a label, enter the
usual keystrokes to program your function, finish with RTN.

LBL A
..
.. ;enter f(x) here
..
RTN


Write a small program:

LBL B
STO 0 ;x
ENTER
x=0? ; if x=0...
e^x ; ...determine h for x=1 to avoid h=0
1E4
/ ; -> h
STO 1
- ;x-h
XEQ A ;f(x-h)
STO 2 ;store in R2
RCL 0 ;x
RCL 1
+ ;x+h
XEQ A ;f(x+h)
RCL 2
- ;f(x+h)-f(x-h)
RCL 1
/
2
/ ;(f(x+h)-f(x-h))/2h
RTN


I didn't try it (no 15C here) but it should work.


So, finally you get:


x XEQ A -> f(x)
x XEQ B -> f'(x) (approx.)


If you want to set h yourself, simply omit the steps from ENTER to STO 1
after LBL B and replace them by a RCL 1 command. In this case it's up to
you to provide a value for h and store it in STO 1.

More information, additional multi-point formulae and some of the
various problems with numerical differentiation can be found on
<http://en.wikipedia.org/wiki/Numerical_differentiation>

Dieter

From: Dieter on 25 Dec 2007 18:35
Mike Dougherty wrote:

> Something my little pea brain has never understood is why, going back
> to the early '80's, the HP 15c was able to preform numerical
> integration, but not numeric differention.
>
> Q1: Does anyone know why??

Probably because it's so trivial that HP didn't bother to integrate it.
Into the 15C, that is. 8-)

> Q2: Is it within the capibilities of these calculators to find the
> numeric derivative of a function, given the proper paramteters?
>
> Q3: Has anyone written a program to perform that operation, and made
> it available? Where?

It's so simple that probably noone would publish it as a proof of his
programming skills:

As you might know, f'(x) = lim (h->0) of (f(x+h)-f(x))/h

For numerical differentiation this expression is used with a small
(but not too small) h in order to approximate the limit, i.e. the
first derivate of f.

Somewhat better precision may be obtained by evaluating the term
(f(x+h)-f(x-h))/2h instead.

h can be provided by the user, or the program itself can set an
appropriate value. A good starting point is h = x/10000 (for x<>0).
The smaller h gets, the better the approximation. OTOH if h gets too
small, the calculator might not be able to distinguish f(x+h) from
f(x-h). Which is just *one* problem here. <8)

Enter f(x) the same way you'd do for integration: use a label, enter the
usual keystrokes to program your function, finish with RTN.

LBL A
..
.. ;enter f(x) here
..
RTN


Write a small program:

LBL B
STO 0 ;x
ENTER
x=0? ; if x=0...
e^x ; ...determine h for x=1 to avoid h=0
1E4
/ ; -> h
STO 1
- ;x-h
XEQ A ;f(x-h)
STO 2 ;store in R2
RCL 0 ;x
RCL 1
+ ;x+h
XEQ A ;f(x+h)
RCL 2
- ;f(x+h)-f(x-h)
RCL 1
/
2
/ ;(f(x+h)-f(x-h))/2h
RTN


I didn't try it (no 15C here) but it should work.


So, finally you get:


x XEQ A -> f(x)
x XEQ B -> f'(x) (approx.)


If you want to set h yourself, simply omit the steps from ENTER to STO 1
after LBL B and replace them by a RCL 1 command. In this case it's up to
you to provide a value for h and store it in STO 1.

More information, additional multi-point formulae and some of the
various problems with numerical differentiation can be found on
<http://en.wikipedia.org/wiki/Numerical_differentiation>

Dieter

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